$T^2 \setminus (\{w\}\times S^1) \text{ is homeomorphic } S^2 \setminus \{ p,q \}$

Question

Let $T^2 = S^1 \times S^1$ be the torus and $p,q \in \mathbb{R}^3, w \in S^1$

Show that $$ T^2 \setminus (\{w\}\times S^1) \text{ is homeomorphic } S^2 \setminus \{ p,q \} $$

Using the stereographic projection, I know that $S^2 \setminus \{ p,q \}$ is homeomorphic to $\mathbb{R}^2\setminus{0}$. How can I use this to proceed here ?

Would appreciate any help.

Answer 1

Hint: you have done the bit that uses the stereographic projection. Now use polar coordinates to show that $\Bbb{R}^2\setminus \{(0, 0)\}$ is homeomorphic to $(0, \infty) \times S^1$ and then show that $(0, \infty)$ is homeomorphic to $S^1 \setminus \{w\}$.